The Gambian President is refusing to accept the end of his leadership following the election of Adama Barrow, a former Argos security guard in London, in December elections.
The country is in a state of emergency and last-ditch diplomatic efforts to convince Mr Jammeh to step aside have failed.
The leaders of Mauritania and Senegal met with the President-elect and Mr Jammeh in the early hours.
But there are no signs the 51-year-old who has ruled the west African country with an iron fist since seizing power in a coup in 1994, is ready to concede defeat.
Nigeria has deployed fighter jets and surveillance planes to Senegal, while Ghana has also pledged to contribute to the regional force.
The Gambia’s army chief has asserted his troops will not fight the entry of outside forces.
“We are not going to involve ourselves militarily. This is a political dispute,” said Ousman Badjie.
“I am not going to involve my soldiers in a stupid fight. I love my men.
“If they (Senegalese) come in, we are here like this,” he added, holding his hands up in the gesture of a surrender.
British tourists have started to arrive back in the UK from The Gambia amid the country’s worsening political crisis.
With fears that the turmoil could see Banjul airport close at short notice, holiday companies urgently put plans into place to fly thousands of Britons home.
Thomas Cook said it had 985 package tourists and 2,500 “flight-only” customers in the West African country and it expects to have brought them all back – on 16 flights – by the end of Friday.
Among those who arrived overnight on Wednesday was Sara Wilkins, from Church Stretton, Shropshire, who said: “We just panicked, just threw everything in a case and just got out of three basically.
“It was a nightmare at the airport – people were crying and panicking. It was just chaos.”
Ebrima Jagne is from The Gambia where he lives with his wife and three-month-old daughter, but works in the UK.
He arrived at Manchester Airport late on Wednesday and told Sky News he had to leave his family behind.
“I’m really desperate… I feel so sad that I have to leave my daughter and my wife and members of my family and I don’t know what’s going to come next.”
Thousands of Gambians have fled across the border to Senegal to escape a possible conflict.
Olof Skoog, president of the UN Security Council, warned “time is running out for diplomacy and mediation to work”.
“I think there is a sense that the whole situation rests within the hands of one person and it is up to that person, the outgoing president of Gambia, to draw the right conclusions and to avoid any further escalation of the situation.”
The Security Council is negotiating a Senegal-backed resolution draft authorising the Economic Community of West African States (ECOWAS) to use “all necessary measures” to remove Mr Jammeh from power.